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Jeans Instability and Gravitational Collapse

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Jeans Instability and Gravitational Collapse Jeans Instability and Gravitational Collapse Gravity introduces a new element into the physics of sound waves (aka stable perturbations) in a gas. Simply put, a gas cloud's self gravity can cause perturbations on sufficiently large scales to become unstable. The goal here is to present a brief overview of fluid dynamics, and to give both rigorous and non-rigorous derivations of the Jeans criterion (named after Sir James Jeans, who first published this analysis in 1902) for gravitational instability. We set the stage by first considering sound waves in a non-gravitating gas, establishing some basic methodology and results. Then we include gravity, and ask how it changes the picture. Fluid Dynamics 101 We consider a fluid in which quantities such as the number and mass densities n(x; t) and ρ(x; t), the pressure P (x; t), the temperature T (x; t), and the velocity field v(x; t) can all sensibly be defined at any point x. As a practical matter, this means that the fluid must be such that the local interparticle spacing ` = n−1=3 is much less than the scale of interest λ. Many (but not all) astrophysical systems fall into this category. The behavior of the fluid is governed by three fundamental equations. First, the continuity equation is a statement of conservation of mass at any point: @ρ + r · (ρv) = 0: (1) @t Integrating this equation over any small volume V surrounding the point x and applying the R divergence theorem to the second term, this equation just says that the rate of change of V ρdV , H the total mass within V , is equal to S ρv · :dS, the rate at which mass flows outward across the surface of V . The second fundamental equation is the Euler equation, which is just the fluid equivalent of Newton's second law: @v rP + v · rv = − : (2) @t ρ The right-hand side is just the pressure gradient, the force per unit volume on the fluid, divided by the density, the mass per unit volume. The left-hand side is the acceleration of the fluid. We can understand the latter statement by considering the rate of change of any quantity Q(x; t) that describes the fluid. As we move with a fluid element, Q changes for two reasons. First, the value of Q at any point x is changing if @Q=@t 6= 0 there. Second, as the fluid moves, the element samples different regions of space, which will have different values of Q if rQ is nonzero. Applying the chain rule for partial differentiation, we have, writing x = (x; y; z), @Q @Q @Q @Q Q(x + δx; y + δy; z + δz; t + δt) − Q(x; y; z; t) ≈ δx + δy + δz + δt; @x @y @z @t so dQ Q(x + δx; y + δy; z + δz; t + δt) − Q(x; y; z; t) ≡ lim dt δ!0 δt @Q @Q @Q @Q = v + v + v + @x x @y y @z z @t @Q = v · rQ + ; @t 1 where v = (vx; vy; vz) = limδt!0 δx/δt. The quantity dQ=dt, the rate of change of Q moving with the fluid, is sometimes called the convective derivative of Q. Setting Q = v, we see that the right hand side of Equation (2) is simply dv=dt. Note that the right-hand side of Equation (2) as written contains only the pressure force. Other forces, such as magnetic fields or internal viscous terms, could also appear there, but we will neglect them here. Equations (1) and (2) represent 4 equations (1 scalar, 1 vector) in 5 unknowns (ρ, P , and v). We need another equation to find a solution. That equation is the equation of state of the fluid P = P (ρ, S) (3) where S is entropy. We will confine ourselves here to isentropic flows, with S = constant. These equations have many interesting solutions, but our interest here is more limited. To start, let's consider a uniform fluid and look at the behavior of small perturbations to that background solution. Specifically, the unperturbed fluid motion is steady, @=@t ≡ 0, with uniform density and pressure ρ0 and P0, and zero velocity v0 = 0. You can easily verify that this solution trivially satisfies Equations (1) and (2). Now let's consider a small perturbation to this baseline solution. We write ρ = ρ0 + ρ1 P = P0 + P1 v = v1 and substitute these into Equations (1) and (2): @ρ @ρ 0 + 1 + r · [(ρ + ρ )v ] = 0 (4) @t @t 0 1 1 @v1 r(P0 + P1) + v1 · rv1 = − : (5) @t ρ0 + ρ1 We then linearize these equations by expanding them to first order, throwing away all products of small terms (having subscript 1). This is the basic mathematical technique employed in all studies of the stability of nonlinear dynamical systems—find an equilibrium solution, then look at the first-order equations describing small perturbations to that solution. (Why? Because we know how to solve linear equations and they give us valuable insight into the physics of the system.) In this case, discarding all products involving more than one small term, and imposing the properties of the unperturbed solution, Equations (4) and (5) become @ρ 1 + ρ r · v = 0 (6) @t 0 1 @v ρ 1 = −∇P : (7) 0 @t 1 Note that the tricky nonlinear v · rv term in the Euler equation, which is the source of a host of complex physical phenomena in real fluids, ranging from turbulent dissipation to shock waves, has disappeared. The key point is that Equations (6) and (7) are soluble. Taking the partial derivative of Equation (6) with respect to time and taking the divergence of Equation (7), and noting that @(r · v1)=@t = r · (@v1=@t), we can eliminate the v1 terms and find @2ρ 1 − r2P = 0: (8) @t2 1 2 We now use the equation of state, Equation (3), to obtain @P @P P1 ≡ δP = δρ = ρ1 ; @ρ S;0 @ρ S;0 so Equation (8) becomes 2 @ ρ1 @P 2 2 − r ρ1 = 0: (9) @t @ρ S;0 We recognize Equation (9) as the wave equation, the solutions to which are traveling waves with wave speed cs, where 2 @P cs = : (10) @ρ S;0 For an ideal gas with P = ρkT=m = Aργ, the adiabatic sound speed is s q γP c = Aγργ−1 = s ρ s γkT = m T 1=2 m −1=2 = 0:91 km=s γ1=2 ; (11) 100 K mH where m is the mean molecular weight. Linearity means that any linear combination of solutions is also a solution. In that case, since any solution can be decomposed into a Fourier series (for a finite region) or a Fourier transform (for an infinite domain), either of which is just a sum of plane-wave solutions of the form ik·x−i!t ρ1 / e (12) (and similarly for other quantities), we can understand the physics of the system by considering the behavior of plane-wave modes. Here ! = 2πf is the angular frequency and k is the wave vector, whose direction is the direction of motion of the plane wave and whose magnitude (the wavenumber) is k = 2π/λ, where λ is the wavelength. Substituting Equation(12) into Equation (9) we obtain the familiar relation between wavenum- ber and frequency: ! = cs k (13) or fλ = cs: (14) Self-gravitating fluids Now let's consider how the above analysis changes when gravity is included. Gravity leaves the continuity equation (1) unchanged, and introduces an additional force into the right-hand side of the Euler equation (2), which now becomes @v rP + v · rv = − − rφ, (15) @t ρ 3 where the gravitational acceleration is a = −∇φ and the gravitational potential φ satisfies Poisson's equation r2φ = 4πGρ. (16) In the spirit of the earlier approach, we seek a steady (@=@t = 0) unperturbed solution with uniform density ρ0 and pressure P0, and zero velocity. We note in passing that this formally leads to a contradiction for the potential: Substituting the \0" quantities into the Euler equation implies 2 rφ0 = 0, so φ0 = constant, but Poisson's equation (16) says r φ > 0. This contradiction is sometimes called the Jeans swindle. Binney & Tremaine (p. 402) discuss some arguments to get around it in practice; see also the article by Kiessling (2003, Adv.Appl.Math. 31, 132; arXiv:astro- ph/9910247v1). In any case, this technical problem with the unperturbed state is not regarded as a serious flaw in the perturbation analysis. We might imagine that other forces, such as rotation or magnetism, are also at play in the initial state, allowing a self-consistent solution to exist. For example, consider a homogeneous self-gravitating fluid of density ρ contained within a rotating cylinder of radius R. The cylinder and the fluid rotate at angular speed Ω about the axis of the cylinder. It is readily shown from Equations (15) and (16) that the fluid can be in true equilibrium, with no pressure gradient (i.e. the gravitational and centrifugal forces balance), if Ω2 = 2πGρ. Ignoring the details of the Jeans swindle and writing ρ = ρ0 + ρ1 P = P0 + P1 v = v1 φ = φ0 + φ1; substituting into Equations (1), (15), and (16), and again using Equation (10) to eliminate P1, we obtain the linearized equations @ρ 1 + ρ r · v = 0 (17) @t 0 1 @v ρ 1 = −c2rρ − rφ : (18) 0 @t s 1 1 2 r φ1 = 4πGρ1 (19) Combining, as before, the time derivative of Equation (17) with the divergence of Equation (18) 2 and employing Equation (19) to eliminate r φ1, we arrive at @2ρ 1 − c2r2ρ = − 4πGρ ρ ; (20) @t2 s 1 0 1 which is identical to Equation (9) apart from the additional term on the right-hand side.
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